cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A353926 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} mu(n)*x^n, where mu = A008683.

Original entry on oeis.org

1, -1, 0, 0, -1, 2, -4, 5, -7, 9, -10, 10, -5, 4, 19, -29, 70, -84, 138, -121, 174, -38, -70, 364, -797, 1423, -2048, 3001, -3479, 3324, -2080, -703, 7923, -15258, 32218, -46063, 68683, -73394, 82847, -45206, -24942, 191996, -422887, 780535, -1225608, 1713171, -2093347
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2022

Keywords

Crossrefs

Cf. A008683, A320781, A353923, A353924, A353925.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, MoebiusMu[n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 47]

A353923 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} tau(n)*x^n, where tau = A000005.

Original entry on oeis.org

1, 2, 0, 3, -1, -1, -1, 9, 1, -7, 0, 4, -1, -19, 2, 94, -2, -57, 2, 81, -4, -186, 3, 226, 3, -632, -2, 1040, 1, -2060, -15, 10975, 17, -7720, -1, 13980, 9, -27595, -18, 50432, -10, -97582, 24, 191827, -17, -364695, 27, 580609, -37, -1338741, 45, 2658068, -11, -4909146, -98
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2022

Keywords

Crossrefs

Cf. A000005, A320779, A328775, A353924, A353925, A353926.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, DivisorSigma[0, n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 55]

Formula

Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} x^n / (1 - x^n).

A353925 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} phi(n)*x^n, where phi = A000010.

Original entry on oeis.org

1, 1, 1, 1, 2, -2, 4, -3, 4, -6, 14, -18, 30, -32, 48, -73, 128, -182, 286, -394, 578, -804, 1386, -1936, 3172, -4298, 7102, -10202, 16414, -23798, 38056, -54997, 86658, -125180, 203396, -292650, 475536, -685314, 1100728, -1619952, 2601166, -3808952, 6114666, -9045400, 14352568
Offset: 1

Views

Author

Ilya Gutkovskiy, May 11 2022

Keywords

Crossrefs

Cf. A000010, A320778, A328774, A353923, A353924, A353926.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, EulerPhi[n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 45]

A353947 Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + Sum_{n>=1} sigma(n)*x^n, where sigma = A000203.

Original entry on oeis.org

1, 2, 1, -1, -3, -1, -1, -3, 3, -17, -1, -6, 3, -22, 1, -28, 1, -68, 7, -262, -13, -199, 27, -341, 29, -647, 3, -1431, -25, -476, -81, -4816, 89, -7384, 637, -17565, -275, -27043, -263, -107113, -453, -98469, -583, -208302, 8121, -371798, -6661, -743344, 3045, -1060666
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2022

Keywords

Crossrefs

Cf. A000203, A320780, A328776, A353924, A353945, A353948, A353949.

Programs

  • Mathematica
    A[m_, n_] := A[m, n] = Which[m == 1, DivisorSigma[1, n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m - 1, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 50]

Formula

Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + Sum_{n>=1} x^n / (1 - x^n)^2.
Showing 1-4 of 4 results.

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